EC 502: Problem Set 4
Due: April 8
1 Investment with Financial Friction
Consider a two-period investment model covered in class. Assume that a firm uses the production function for output Yt = F(At
, Kt
, Lt) given capital Kt
, labor Lt
, and productivity At
. The function is given by
The firms also face the following capital adjustment costs:
1. If the firm faces the wage rate W per unit of labor L, then write down an expression for the variable profit function Π(K; A, W) defined by
2. The firm must choose its period 0 investment level I0 to maximize the present discounted value of dividends
where D0 = Π(K0; A0, W0) − I0 − Φ(I0, K0), D1 = Π(K1; A1, W1), and K1 = I0 + (1 − δ)K0. Derive the optimality condition for investment I0. Interpret this condition, in words.
3. Given optimal investment choices, how does the optimal investment I0 respond to an in-crease in r, W0, A0, W1, and A1, respectively?
4. Now suppose there is financial friction in the economy, and firms cannot investment more than their current profits, D0 ≥ 0. Under what conditions does this financial constraint bind?
5. If the above financial constraint binds, how does the optimal investment I0 respond to an increase in r, W0, A0, W1, and A1, respectively?
2 Labor Income Taxes
Consider an RBC economy similar to the one described in the lecture note with two periods, but we assume capital is fixed (there is no investment). We will introduce labor income tax to understand whether the changes in labor income tax can generate business cycles.
Households. Households have a utility function in each period given by U(Ct
, lt), where Ct is consumption in period t and lt
is labor supply in period t. Households may save the amount A0 in period 0 with a given interest rate r. Labor in period t earns wages at rate Wt
, but the labor income is taxed at marginal rate τLt ≥ 0 in period t, so that the households only receive the effective wage (1 − τLt)Wt
for each unit of labor supplied. The government rebates the taxes back to the households in lump-sum form. with transfers Qt = τLtWtLt
in each period. The utility maximization problem is given by
Assume that household preferences are given by
with ν > 0.
Firms. Firms have a production function given by for t = 1, 2. K is the fixed capital stock in this economy, and there is no investment. Firms face the wage rate Wt and solve the static profit maximization problem
The resulting optimal choice of labor for firms, Lt
, represents labor demand in period t.
General Equilibrium. General equilibrium in this economy is a set of prices and quantities Wt
, r, Ct
, Yt
, lt
, and Lt such that
• Households optimize their utility as laid out above given Wt and r
• Firms maximize profits as laid out above given Wt
• Markets Clear
– Labor Markets Clear: Lt = lt
– Resource Constraints Hold: Yt = Ct
1. Derive the household optimality conditions for the optimal labor choices lt
in periods t = 0, 1. These are also known as the labor supply curves in each period.
2. Using 1) the resource constraint, 2) the production function, and 3) labor market clearing, write the household labor supply curve as an equation with Wt on the left hand side and some function of At
, Lt
, and K on the right hand side.
3. Derive the firm’s optimality conditions for labor Nt
in each period t = 0, 1. These are also known as the labor demand curves.
4. For period t, set the household labor supply curve equal to the firm labor demand curve and eliminate Wt
from this equation to derive a unified labor market equilibrium condition.
5. If τN0 increases, how do Lt
, Yt and Ct redpond?
6. If τN0 and τN1 increase by the same amount, how do Lt
, Yt and Ct redpond?